If a parked car door is hit by another car and the owner has the chance to get it fixed cheaply but the other party is denying responsibility can the car still be fixed cheaply by the owner?

Well you own the car so you can do whatever you want. However, it may be in your best interest to let the insurance companies handle it. Remember, no matter how careful you can be, scratches and even minor dents and dings can add up, and have somewhat of an impact on the resale value of you vehicle. If the damage is somewhat minor but you'd still like to get it fixed and a small claims lawsuit is not really in your best interest at this time (it costs time, money to file, time to recoup payment, etc.), it may really be in your best interest to let insurance handle it.